2010, 17: 90-103. doi: 10.3934/era.2010.17.90

Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces

1. 

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States

Received  June 2010 Published  October 2010

On reflexive spaces trigonometrically well-bounded operators (abbreviated "twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates "transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.
Citation: Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90
References:
[1]

D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123. doi: 10.1017/S0305004100055559.

[2]

B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982.

[3]

E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715. doi: 10.1007/BF01200555.

[4]

E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.

[5]

E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789. doi: 10.1007/BF01202516.

[6]

E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170. doi: 10.1016/0022-1236(87)90128-5.

[7]

E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.

[8]

E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189. doi: 10.1090/S0002-9947-97-01896-5.

[9]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132. doi: 10.1215/S0012-7094-97-08804-9.

[10]

E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133. doi: 10.1023/A:1009794510984.

[11]

E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61. doi: 10.4064/sm144-1-2.

[12]

E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.

[13]

E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.

[14]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177. doi: 10.1007/s00041-004-8009-z.

[15]

E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517. doi: 10.1112/plms/s3-53.3.489.

[16]

D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007.

[17]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[18]

M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.

[19]

P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.

[20]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.

[22]

G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.

[23]

R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.

[24]

B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.

[25]

G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975.

[26]

J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679. doi: 10.1216/rmjm/1181069753.

[27]

J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.

[28]

P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.

[29]

S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308. doi: 10.1006/jfan.1996.2986.

[30]

L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.

[31]

A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959.

show all references

References:
[1]

D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123. doi: 10.1017/S0305004100055559.

[2]

B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982.

[3]

E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715. doi: 10.1007/BF01200555.

[4]

E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.

[5]

E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789. doi: 10.1007/BF01202516.

[6]

E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170. doi: 10.1016/0022-1236(87)90128-5.

[7]

E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.

[8]

E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189. doi: 10.1090/S0002-9947-97-01896-5.

[9]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132. doi: 10.1215/S0012-7094-97-08804-9.

[10]

E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133. doi: 10.1023/A:1009794510984.

[11]

E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61. doi: 10.4064/sm144-1-2.

[12]

E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.

[13]

E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.

[14]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177. doi: 10.1007/s00041-004-8009-z.

[15]

E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517. doi: 10.1112/plms/s3-53.3.489.

[16]

D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007.

[17]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[18]

M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.

[19]

P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.

[20]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.

[22]

G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.

[23]

R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.

[24]

B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.

[25]

G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975.

[26]

J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679. doi: 10.1216/rmjm/1181069753.

[27]

J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.

[28]

P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.

[29]

S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308. doi: 10.1006/jfan.1996.2986.

[30]

L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.

[31]

A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959.

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